Years and Authors of Summarized Original Work
2004; Elkin, Peleg
Problem Definition
For a pair of numbers \( { \alpha,\beta } \), \( { \alpha \ge 1 } \), \( { \beta \ge 0 } \), a subgraph \( { G^{\prime} = (V,H) } \) of an unweighted undirected graph \( { G = (V,E) } \), \( { H \subseteq E } \), is an \( { (\alpha,\beta) } \)-spanner of G if for every pair of vertices \( { u,w \in V } \), \( \text{dist}_{G^{\prime}}(u,w) \le \alpha \cdot \text{dist}_G(u,w) + \beta \), where \( { \text{dist}_G(u,w) } \) stands for the distance between u and w in G. It is desirable to show that for every n-vertex graph there exists a sparse \( { (\alpha,\beta) } \)-spanner with as small values of α and β as possible. The problem is to determine asymptotic tradeoffs between α and β on one hand, and the sparsity of the spanner on the other.
Key Results
The main result of Elkin and Peleg [8] establishes the existence and efficient constructibility of \( { (1+\epsilon,\beta) } \)-spanners of size \( {...
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Elkin, M. (2016). Sparse Graph Spanners. In: Kao, MY. (eds) Encyclopedia of Algorithms. Springer, New York, NY. https://doi.org/10.1007/978-1-4939-2864-4_387
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